Microscopic modeling of nonlinear transport in ballistic nanodevices

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Microscopic modeling of nonlinear transport in ballistic nanodevices
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  IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 50, NO. 9, SEPTEMBER 2003 1897 Microscopic Modeling of Nonlinear Transport inBallistic Nanodevices Javier Mateos, B. G. Vasallo, Daniel Pardo, Tomás González, Jean-Sébastien Galloo, Sylvain Bollaert,Yannick Roelens, and Alain Cappy  Abstract— By using a semi-classical two-dimensional (2-D)Monte Carlo simulation, simple ballistic devices based onAlInAs/InGaAs channels are analyzed. Our simulations quali-tatively reproduce the experimental results in T- and Y-branch junctions as well as in a ballistic rectifier appearing as a result of electron ballistic transport. We show that a quantum descriptionof electron transport is not essential for the physical explanationof these results since phase coherence plays no significant role.On the contrary, its srcin can be purely classical: the presence of classical electron transport and space charge inside the structures.  Index Terms— Ballistic transport, Monte Carlo simulation, tera-hertz devices. I. I NTRODUCTION O NE of the possible approaches for overcoming the limitsof traditional scaling when reaching the nanometerrange (which has been the main engine of the progress of thesemiconductor industry) is the use of devices exploiting theballistic transport of electrons. Ballistic devices fabricatedusing the GaAs/AlGaAs heterojunction operating at lowtemperature have been demonstrated [1], [2]. However, recent works have achieved an important improvement using InGaAschannels with high In content [3]–[6], room temperature operation is possible since the mean-free-path of electrons isstill larger than 100 nm, which is a feature size in the reachof current lithographic techniques. The small size of theseballistic devices and the high velocity of electrons inside reducesignificantly their transit time, and as a result, the fabricationof devices for data processing at ultra-high bit rate can beenvisaged [4], [5]. Moreover, InGaAs-based ballistic devices offer the advantage of being compatible with modern HEMTtechnology; indeed, AlInAs/InGaAs HEMTs nowadays operatein the millimeter and submillimeter wave frequency range [7].Thus, the integration of ballistic devices with HEMTs in orderto benefit from their complementary advantages and reach theterahertz regime appears feasible in the near future. Manuscript received October 22, 2002; revised May 23, 2003. This work wassupported in part by the European Commission through the NANOTERA underProject IST-2001-32517, by the Dirección General de Investigación (Ministeriode Ciencia y Tecnología) and FEDER under Project TIC2001-1754, and by theConsejería de Cultura de la Junta de Castilla y León under Project SA057/02.The review of this paper was arranged by Editor H. Sakaki.J. Mateos, B. G. Vasallo, D. Pardo, and T. González are with the Universidadde Salamanca, Salamanca, Spain (e-mail: javierm@usal.es).J.-S Galloo, S. Bollaert, Y. Roelens, and A. Cappy are with the Institutd’Electronique de Microélectronique et de Nanotechnologie U.M.R. C.N.R.S.,Département Hyperfréquences et Semiconducteurs, Villeneuve D’Ascq Cédex,France.Digital Object Identifier 10.1109/TED.2003.815858 The first step in the design of ballistic structures is the de-termination of their optimal geometry. At this level, simulationtools constitute a valuable alternative to the expensiveand time-consuming test-and-error procedure. Some theoretical descrip-tions of the operation of ballistic devices have been proposed[8]–[10], [19], always starting from a coherent transport de- scription based on the Landauer–Buttiker formalism [11], [12]. In this paper, we present a microscopic analysis, performed bymeans of Monte Carlo (MC) simulations, of the transport prop-erties of several structures based on AlInAs/InGaAs ballisticchannels specially designed to be applied in electronic devicesforterahertzdataprocessing.MCsimulationsprovideaninsightof the processes taking place inside the devices, thus allowingus to relate the macroscopic results of the experiments with themicroscopic behavior of electrons. Our model, which is basedon a semiclassical transport description, is able to qualitativelyreproduce the main features of the ballistic effects measured inbasic devices like T-branch (TBJs) [6] and Y-branch (YBJs) junctions [1], [3] and ballistic rectifiers [2], [4], thus demon- strating that coherent transport plays no significant role on themain characteristics of these devices.In Section II, the details about the MC model and the simu-lated structures will be given. In Section III, the validity of ourapproachwillbecheckedbycomparisonofthesimulationswithmeasurementsperformed inrealAlInAs/GaInAschannels,withthe layer structure typically used in the fabrication of HEMTs.Then, in Section IV, simulations of ballistic TBJs, YBJs, andballistic rectifiers will be presented, qualitatively reproducingthe main experimental findings shown in the literature [1]–[4], [6].II. M ONTE  C ARLO  M ODEL We make use of a semiclassical ensemble MC simulator self-consistently coupled with a 2-D Poisson solver. The transportmodel locally takes into account the effect of degeneracy andelectron heating by using the rejection technique and the self-consistent calculation of the local electronic temperature andFermi level [13]. The surface charges appearing at the bound-aries of the semiconductors in contact with dielectrics are alsoconsidered in the model [14]. The validity of this approach hasbeen checked in previous works by means of the comparisonwith experimental results of static characteristics, small signalbehavior, and noise performance of a 0.1- m gate AlInAs/In-GaAs lattice matched HEMT (InP based) [14], [15]. Since con-tactinjection isacriticalpointwhen dealingwithballistictrans-port, the velocity distribution and time statistics of injected car-riers will be accurately modeled [16], [20]. 0018-9383/03$17.00 © 2003 IEEE  1898 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 50, NO. 9, SEPTEMBER 2003 Fig. 1. Three-dimensional (3-D) geometry and layer structure of the ballistic channels and scheme of the 2-D front-view (FV) and top view (TV) Monte Carlosimulations. The main features of the ballistic channels that will besimulated are shown in Fig. 1, where the geometry is usuallydefined in the fabrication process by a typical mesa etching.The real layer structure fabricated by molecular beam epitaxy(MBE) consists of an InP substrate, a 200-nm Al In Asbuffer followed by a 15-nm-thick In Ga As strainedchannel, three layers of Al In As (a 5-nm spacer, a-doped layer, and a 10-nm Schottky layer), and finally, a15–nm-thick In Ga As cap layer cm .Layers with two different -dopings have been fabricated (4and cm ), and they will be modeled as a 4-nmlayer doped at cm for the cm andcm for the cm (moreover, astructure with cm will be also simulated usingcm ). To account for the thickness of the-doped layer while keeping the overall size, the dimensionof the spacer and Schottky layers are set to 3 and 8 nm,respectively.For the correct modeling of these devices, a 3-D simulationwould be necessary in order to take into account the effect of the lateral surface charges and the real geometry of the struc-tures. However,for themoment, only a 2-DMC model has beendeveloped, and some simplifications and assumptions must bemade. Indeed, two different types of 2-D simulations will beperformed: front-view (FV) and top-view (TV). Within the FVsimulations, the layer structure will be taken into account, butthe device in the dimension is considered to be homogeneous.This kind of simulations will be useful for simple structures,like homogeneous channels, and will provide the concentrationof carriers in each layer. On the other hand, to account for thetopgeometryofmorecomplicateddevices(suchasTBJs,YBJs,or ballisticdiodes),TV simulationswillbecarriedout.Theyareperformed in the plane; therefore, the real layer structure isnot included, and only the InGaAs channel will be simulated. Inordertoaccountforthefixedpositivechargesofthewholelayerstructure, a net doping is assigned to the channel in TV simu-lations, but impurity scattering is switched off. In this way, theelectron transport through the undoped channel is well repro-ducedsincethisisa“virtual”dopingassociatedwiththechargesofthecapand -dopedlayers.Ontheotherhand, anegativesur-face charge density is assigned to the semiconductor-air inter-facestoaccountfortheinfluenceofthesurfacestatesoriginatedby the etching processes. Therefore, to ensure the accuracy of this TV approximation, the values of two important parametersmust be carefully chosen: the background doping in the channeland the lateral surface charge density .III. B ALLISTIC  C HANNELS Initially, the validity of the simulation tools is checked bycomparison of the numerical results with experimental Hall-ef-fect measurements of carrier concentration and mobility of twofabricated layers ( cm and cm ).To improve the ballistic character of electron motion, impurityscatteringwillbediminishedbyallowingtransportonlythroughthe InGaAs channel. For this sake, recessed channels (the caplayer is removed) will be used. By adjusting the surface chargeat the cap layer to a value of cm , andat the bottom of the recess (free AlInAs interface) tocm , we have obtained a good agreement with themeasured values of Hall density in both recessed and nonre-cessed layers. However, we have appreciated that the channelmobility obtained by MC simulations overestimates the mea-sured values (around 10000 cm Vs). We have checked thatthis discrepancy is due to remote impurity scattering (not in-cluded in the MC model) since the experimental mobility of the channel is improved when a thicker spacer layer is growth,reaching a better agreement with the MC values (around 14000cm Vs).We have performed the FV simulation of two different chan-nels (considering the whole layer structure) with lengthsnm and nm between two ohmic contacts. Thediffusive or ballistic character of transport depending on canbe monitored, from the point of view of the Monte Carlo simu-lation, by means of the number of scattering processes the elec-trons overcome while crossing the sample. In practice, trans-  MATEOS  et al. : MICROSCOPIC MODELING OF NONLINEAR TRANSPORT IN BALLISTIC NANODEVICES 1899 Fig. 2. Normalized current versus applied voltage for nonrecessed channelswith lengths of 100 nm (dotted lines) and 1000 nm (solid lines) and different   dopings. The values obtained from TV simulations with        cmand      nm are also shown. The inset shows the normalized saturationcurrent as a function of the channel length for     2     cm . port will never be completely ballistic since, even for very short, electrons always overcome a few scattering mechanisms.Some other features can give us information about how bal-listic the transport inside the channels is, ranging from diffu-sive (for the channel with nm) to quasiballisticnm . One of these indicators is the ratio between the satu-ration current and the total current injected by the contacts, which is the maximum current that may flow through thechannels [16], [20]. In the case of completely ballistic transport, since every electron injected by the cathode arrivesat the anode (for high enough biasing). When scattering mech-anisms appear, some carriers return to the anode, making theratio lower than unity. In Fig. 2, is representedas a function of the applied voltage, showing that in-creases when reducing due to the lower amount of scatteringmechanisms, which leads to a more pronounced velocity over-shoot srcinated by the enhanced ballistic transport. The valueof as a function of is plotted in the inset of Fig. 2,showing that for nm, exceeds 95%, and elec-tron transport can be considered to be quasiballistic. We haveto note that the results plotted in Fig. 2 correspond to nonre-cessedchannels(includingthecaplayer).Inthecaseofrecessedchannels, impurity scattering in the cap layer is prevented, andthe ratio is increased but only for the long channels.The values of for the 100-nm nonrecessed and recessedchannels overlap since the mean free path associated with ion-ized impurity scattering is longer than the length of the struc-tures. Therefore, the recessed technology will be useful for im-proving the ballistic transport only when the length of the chan-nels approaches the ballistic/diffusive limit (around 200 nm).The simulation of the ballistic channels can also be madewithin the TV approach, considering a background dopingwith no impurity scattering. In Fig. 2, the inductance–voltage( – ) curve obtained with this model is compared with the cor-rect FV simulations of the 100-nm channels. In this case, tocompare with the FV results, no lateral surface charge is con-sidered since this charge can not be accounted for in FV simu- Fig.3. (a)Electricpotentialatthebottomofthecentralbranch(opencircuited)of the TBJ and (b) normalized horizontal current when biasing the left andright contacts in push-pull fashion          0      for different valuesof the lateral surface charge. The inset shows the geometry of the TBJ with50-nm-wide and 75-nm-long branches. lations. It can be appreciated that setting the background dopingtoavalue cm andconsidering injectingcontactswith cm [16], [20], TV simulations satisfac- torily reproduce the behavior of the ballistic channels (both re-cessed and nonrecessed). These values will therefore be used inall further TV simulations.IV. B ASIC  D EVICES  A. T-Branch Junctions As a first example of devices that benefit from ballistic trans-port, we will analyze the main features of a TBJ. In [6], the ex-perimental measurements of the negative potential generated atthe bottom of the central branch (open circuited, ) of aTBJ ( ) when the left and right ones are biased in push-pullfashion ( ) are presented. As stated in [10]and [19], this property of the three terminal ballistic junctions(both TBJs and YBJs) can be very useful from a practical pointof view since it can be exploited to perform logical operations[17], second harmonic generation [5], [18] or rectification [2], [4].In Fig. 3(a), we show the values of calculated withTV-MC simulations of the TBJ sketched in the inset consid-ering different values for the lateral surface charge density .The value of at equilibrium has been subtracted from the re-sults to account for the difference of electrochemical potentials.The simulations do qualitatively reproduce the experimental  1900 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 50, NO. 9, SEPTEMBER 2003 Fig. 4. (a) Electron concentration and (b) electric potential profiles along themiddle of the horizontalbranch of the TBJ ofFig. 3 with          2     cmfor different bias conditions          0      . The insets of (b) show thevertical potential profile in the middle of the central branch for several biasings,and the values of the potential at the bottom of this branch    and at the centerof the junction    as a function of     . results, thus also supporting the validity of our TV-MC model.As seen in Fig. 3, the values of depend on the amount of surface charge considered at the semiconductor-air interfacessince it controls the intensity of space charge effects [theamplitude of the potential minimum observed in Fig. 4(b)]. Theeffect of the surface charge is similar to that observed in [6]by changing the potential of a top gate contact, both affectingthe Fermi level pinning and the electron concentration insidethe TBJ. The correct values of the background doping andsurface charge density to be used in the TV-MC simulation willhave to be further adjusted, taking as a base the experimentalmeasurements of channels with different length and widththat we plan to fabricate. Nevertheless, as pointed out in [4],the lateral depletion length of InGaAs channels is about10–30 nm, which corresponds to surface charge densities of cm (using cm ), which isin reasonable agreement with the range of values used in oursimulations.The negative values of are related to space-charge effectssrcinated by the joint action of the surface charge at the semi-conductor-air interfaces, the background positive fixed charge, and the inhomogeneous charge distribution associatedwith the ballistic motion of carriers injected at the contacts. Thesurface charge lowers the electric potential when moving awayfrom the contacts, provoking the progressive depletion of thechannel, thus leading to the typical minimum of potential andconcentration (see Fig. 4) characteristic of space-charge limited Fig. 5. Electric potential at different positions of the horizontal channel of theTBJ of Fig. 3 as a function of the biasing          0      . ballistic conditions [16], [20]. At equilibrium, this minimum is in the middle of the structure. When the TBJ is biased, thecarrier number inside the device increases since more and moreelectrons are able to surmount the potential barrier (i.e., moremodes are opened under the Landauer–Buttiker description[8]–[10], [19]), leading to the nonlinear – characteristic shown in Fig. 3(b). Moreover, the concentration exhibits anasymmetric shape (higher near the negative electrode due tothe electron ballistic motion) so that a shift of the potentialminimum toward the negative electrode takes place. As aconsequence, the potential at the center of the horizontalchannel is always lower than the equilibriumvalue and decreases with larger . The inset of Fig. 4(b) showsthe vertical profile of the electric potential in the middle of thecentral branch. It can be observed that the variations of versus propagate down to the bottom of the vertical branch,thus srcinating the characteristic bell-shaped values of .Therefore, the vertical branch acts just like a voltage probeconnected to the horizontal channel, detecting the potentialvariations at the junction. This happens because the penetrationof carriers in the central branch is just the consequence of thenonzero vertical velocity component of the carriers flowingwithin the horizontal channel, thus being almost independentof . In the case that the vertical branch is slightly shiftedfrom the center of the horizontal channel, the potential at the junction , and therefore , are no longer symmetric,taking positive values for some values of (Fig. 5), as alreadypredicted in [10] and [19] for asymmetric junctions. In Fig. 3, we can also observe that the negative values of the versus curve reach a maximum for an intermediatevalue of , just when the width of the channel coincides withthe lateral depletion induced by the surface charge (forcm , nm, and the theoretical effec-tive width of the channel becomes 0). InFig. 6(a), it is shown that under these conditions, and for lowvalues of , follows the predicted quadratic dependence on[10], [19], but it becomes linear for high . In Fig. 6(b), the
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